This paper opens the study of quasi-isometric embeddings of symmetric
spaces. The main focus is on the case of equal and higher rank. In this
context some expected rigidity survives, but some surprising examples also
exist. In particular there exist quasi-isometric embeddings between spaces
and
where there is no
isometric embedding of
into
.
A key ingredient in our proofs of rigidity results is a direct generalization
of the Mostow–Morse lemma in higher rank. Typically this lemma is
replaced by the
quasiflat theorem, which says that the maximal quasiflat
is within bounded distance of a finite union of flats. We improve this by
showing that the
quasiflat is in fact flat off of a subset of codimension
.
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